Me338a Continuum Mechanics

Home , Conservation of mass, Continuity equation, Solid mechanics

global vs local balance equations The balance equations of continuum mechanics serve as a basic set of equations required to solve an initial boundary value problem of thermomechanics for the primary vari- ables. This section is devoted to derivation of the fundamen- tal balance laws of continuum thermomechanics. In what P ME338A follows, we consider a certain volume closed by the boundary ∂P. For this part of the body, we balance a volumetric CONTINUUM MECHANICS source and a surface ﬂux with the temporal change of the quantity for which the balance principle is constructed. Ini- tially, the balance equation is derived on the entire subset P, i.e., in a global form. In order to derive the local form of a lecture notes 11 balance law, we transform the surface ﬂux terms into a vol- tuesday, may 06, 2008 ume term through Gauss’ integral theorem and localize the resulting expression to any arbitrary point x ∈PS provided that the continuity conditions are met.

T˜ n N t da X dA F x − PB t F PS ∂P ∂PB S B S

material vs spatial balance equations The global and local material balance equations valid on the reference conﬁguration PB and at point X ∈PB can be re- cast into their spatial format valid on the current conﬁgura- tion PS and at point x ∈PS, respectively.

88 3 Balance equations 3 Balance equations

3.3 Balance of mass local balance of mass / differential form modiﬁcation of rate terms,localization to any point inside P total mass m of a body P d d ρ0 dV = J ρt dV = 0 (3.3.8) m = ρ0 dV = ρt dv (3.3.1) PB dt PB dt PB PS local balance of mass, material version, in terms of ρ0 in terms of material and spatial mass density ρ0 and ρt from transformation of volume elements dv = J dV with d ρ dv = J ρ dV = ρ dV thus ρ = 0 (3.3.9) PS t PB t PB 0 dt 0 ρ0 = J ρt (3.3.2) conservation of mass solid mechanics mass exchange of body P with its environment through mass ﬂux across the surface ∂P “The mass in a material body does not change.”

sur alternative statement from global spatial version m = 0 (3.3.3) d ( ρ )= d ρ + ρ d and through volume source within P J t J t t J dt dt dt mvol = 0 (3.3.4) d = J ρt + ρtdiv(v) (3.3.10) dt global balance of mass / integral form ∂ = ρ + (ρ ) = J ∂ t div t v 0 “The time rate of change of the total mass m of a body P is t balanced with the mass exchange due to the contact mass with the following transformations sur vol − ﬂux m and the at-a-distance mass exchange m .” dJ/dt = J F t : F˙ = J div(v) d ρ = ∂ρ ∂ + ∇ ρ · m = msur + mvol (3.3.5) d t/dt t/ t x t v dt ρtdiv(v)+∇xρt · v = div(ρt v) global balance of mass, material version, i.e., in terms of ρ0 local balance of mass, spatial version, in terms of ρt d ρ0 dV = 0 (3.3.6) dt PB d ∂ ρ ρ + ρ div(v)=0 ρ + div(ρ v)=0 (3.3.11) global balance of mass, spatial version, i.e., in terms of t t t ∂ t t dt t d ρ = J t dV 0 (3.3.7) continuity equation dt PB Euler 1757 ﬂuid mechanics 89 90 3 Balance equations 3 Balance equations remarks summary of useful material vs spatial transformations

◦ balance equations can be phrased in a global / integral or −1 ·{}· F N n in a local / differential format, both combinations can either N {♦}·n be formulated in terms of material quantities or in terms of da spatial quantities, the particular choice of the balance equa- {◦} {◦} 0 dA F t − tion depends on the application PB t F PS ∂P ∂PB S ◦ for descriptions in fixed domains PB, the conservation of B S mass dρ0 /dt = 0 is usually fulfilled automatically, it con- sists of only one term, the time rate of change of the balance material vs spatial volume terms Piola transform quantity within the fixed domain PB, these formulations are {◦} {◦} global comparison of volume terms 0 and t typical in solid mechanics {◦}0 dV = {◦}t dv (3.3.12) PB PS ◦ for descriptions in moving domains PS, the conservation with volume transformation dv = J dV of mass ∂ρt / ∂t + div(ρt v)=0 is referred to as the conti- local comparison of volume terms {◦}0 and {◦}t nuity equation, it nicely illustrates that for time derivatives in moving domains, we have a time evolution term inside {◦} = {◦} the moving domain ∂ρt / ∂t and a convective term div(ρt v) 0 J t (3.3.13) accounting for the in- or outflux through the moving boundary, these formulations are common in fluid mechanics material vs spatial surface terms Piola transform ◦ {} {♦} terms between the material and spatial formulation can global comparison of surface terms and Piola transforms be transformed into one another through the {}·dA = {♦}·da (3.3.14) and Reynold’s transport theorem ∂PB ∂PS with area transformation da = J F−1 · dA, Nanson’s formula ◦ balance equations have a somewhat “hierarchical order”, local comparison of surface terms {} and {♦} “lower order” balance equations can be used to simplify “higher order” balance equations, e.g., we will see that the {} = {♦}· −t {} = {♦} balance of mass can be used to simplify the balance of linear J F Div J div (3.3.15) momentum

91 92 3 Balance equations 3 Balance equations material vs spatial time derivative 3.4 Balance of linear momentum material time derivative: @fixed material position X P ∂ total linear momentum p of a body d {◦} = {◦} = {◦}˙ : ∂ (3.3.16) = ρ = ρ dt t X fixed p 0 v dV t v dv (3.4.1) PB PS spatial time derivative: @fixed spatial position x P momentum exchange of body with environment through sur ∂ ∂ contact forces f {◦} := {◦} (3.3.17) ∂t ∂t sur x fixed f = T d A = t d a (3.4.2) ∂P ∂P Euler theorem B S f vol local comparison of time derivatives and at-a-distance forces vol d ∂ f = ρ0 b dV = ρt b dv (3.4.3) {◦} = {◦} + ∇ {◦} · v (3.3.18) PB PS dt ∂t x in terms of contact/surface forces T = P · N and t = σ · n Reynold’s transport theorem and volume forces b global comparison of time derivatives d ∂ {◦} = {◦} + {◦} ⊗ · T˜ n dV ∂ dv v n da (3.3.19) dt PS PS t ∂PS N t local comparison of time derivatives da X dA F x − ∂ PB t d F PS ∂P {◦} = J {◦} + div({◦} ⊗ v) (3.3.20) ∂PB S dt 0 ∂t t t B S

Reynold’s transport theorem global balance of momentum / integral form

“The rate of change of the quantity {◦}0 in a fixed material “The time rate of change of the total momentum p of a body volume PB equals the rate of change of the quantity {◦}t in P is balanced with the momentum exchange due to contact a fixed spatial control volume PS plus the flux through the momentum flux / surface force f sur and the at-a-distance boundary of the control domain ∂PS.” momentum exchange / volume force f vol.” d p = f sur + f vol (3.4.4) dt 93 94 3 Balance equations 3 Balance equations global balance of momentum, material version, in ρ0 & T Cauchy’s first law of motion, Cauchy [1827] d ρ0 v dV = T dA + ρ0 b dV (3.4.5) P ∂P P t dt B B B ρ0 a = Div(P )+ρ0 b (3.4.11) global balance of momentum, spatial version, in ρt & t d equilibrium equation solid mechanics J ρt v dV = t da + ρt b dv (3.4.6) dt PB ∂PS PS Piola transforms local balance of momentum / differential form where {◦}0 = ρ0 and {◦}t = ρt,and{} = P and {♦} = σ modification of rate terms dp /dt ρ v = J ρ v 0 t d d d = σ · −t ( t)= (σt) p = (ρ0 v) dV = (J ρt v) dV (3.4.7) P J F Div P J div dt PB dt PB dt ρ0 b = J ρt v modification of surface terms f sur Reynold’s transport theorem sur Cauchy Gauss t {◦} = ρ {◦} = ρ f = T dA = P · N dA = Div(P ) dV where 0 0v and t tv ∂PB ∂PB PB ∂ = Cauchy= σ · Gauss= (σt) d t da n da div dv (ρ0v)=J (ρtv)+div(ρtv ⊗ v) (3.4.12) ∂PS ∂PS PS dt ∂ (3.4.8) local balance of momentum, spatial version, in ρt & σ thus d t ρ = (σt)+ρ (ρ0 v) dV = Div(P )+ρ0 b dV t a div t b PB dt PB ∂ (3.4.13) (3.4.9) (ρ )= (σt − ρ ⊗ )+ρ d ( ρ ) = (σt)+ ρ t v div t v v t b J t v dV J div J t b dV ∂t PB dt PB equilibrium equation solid & fluid mechanics local balance of momentum, material version, in ρ0 & P d (ρ v)=Div(Pt)+ρ b (3.4.10) dt 0 0 the balance of momentum is maybe the most important equation in solid and fluid mechanics, again, it can be phrased reduction by subtracting weighted version of balance of mass globally or locally, on fixed and moving domains v dρ0/dt = 0 and with a = dv /dt

95 96 3 Balance equations from global to local (i) modification of rate terms transform all integral terms to fixed domain PB (ii) modification of surface terms transform all boundary flux terms on ∂PB into volume terms in PB with Green/Gauss theorem (iii) localize the global version for each point in P from material to spatial (i) apply Piola transform for volume terms

{◦}0 = J {◦}t (ii) apply Piola transform for surface terms Div{} = J div{♦} (iii) apply Reynold’s transport theorem for time derivatives d {◦} = ∂ {◦} + ({◦} ⊗ ) dt 0 J ∂t t div t v for quasi-static problems, e.g., in the static analysis of struc- tures or for material testing, the acceleration term is usually neglected, i.e., a ≈ 0, such that the equilibrium equation would reduce to

t t Div(P )+ρ0 b = 0 div(σ )+ρt b = 0 for vanishing volume forces as common to most applications, i.e., b = 0, the equilibrium equation would then reduce to

Div(Pt)=0 div(σt)=0 a typical example of a volume force term would be gravity, however, in most analyses this time is typically neglected